In a letter to investors this month, Mr. Ackman disclosed that through the end of November, the fund had declined 13.5 percent this year after accounting for fees. (Pershing Square Holdings shouldn’t be confused with Pershing Square L.P., Mr. Ackman’s hedge fund, although the two vehicles have the same investment strategy.)

That’s obviously a big disappointment, considering the Standard & Poor’s 500-stock index was up 7.6 percent over the same period. But that’s not what some big investors were complaining about to me this week.

In the same letter, Mr. Ackman reported that during the nearly four years since it began, Pershing Square Holdings had gained a total of 20.5 percent. That would be considered mediocre at best, considering the S.&P. 500 gained over 67 percent during the same period.

That means Pershing Square kept approximately 72 percent of the fund’s gains for itself, leaving investors with the measly remains.

A spokesman for Mr. Ackman declined to comment. But the reality is that many hedge funds, not just Mr. Ackman’s, reap far higher percentages of their gains than that stated in their fee structure. That’s because when they experience substantial losses — as Pershing Square did last year and is on track to do this year (its year-to-date loss through Tuesday was 12.4 percent) — they don’t have to give anything back.

And for many hedge funds the results are even worse. Most hedge funds charge the proverbial two-and-20 — 2 percent of assets under management and 20 percent of any gains above a certain threshold. By these measures, Pershing Square Holdings’ lower fee structure is a relative bargain.

How could Pershing Square have kept 72 percent of the gains, given that its fee structure calls for a performance fee of just 16 percent?

The answer can be found in relatively simple math. As a simple example, consider an investment of $1 million in a fund that generates a 10 percent return in years one and two and then loses 5 percent in years three and four. The investor would end up with about $1.09 million, a total gain of $90,000, or 9 percent, over the four years before fees.

But now consider the return after deducting a 20 percent performance fee. In years one and two, the fund manager earns $20,000 and $20,400 for a total of $40,400. The fund’s manager earns nothing in years three and four. After deducting the fees, the investor would end up after the four years with just $1.05 million, a total return of 5 percent. But the $40,400 earned by the fund is nearly 45 percent of the investor’s total gains before fees — not 20 percent. (And that’s not even figuring in a 1.5 or 2 percent management fee.)

If the losses are big enough, the hedge fund manager can capture 100 percent of the gross return, or investors can lose money even as fund managers line their pockets.

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Because of concerns over high fees and disappointing results, some endowments and pension funds, including those in Illinois, New Jersey and Rhode Island, have cut back substantially on their hedge fund allocations this year, following the lead of Calpers, the largest pension fund in the United States, which said in 2014 that it would exit hedge funds entirely.

Through the third quarter of this year, investors had withdrawn about $51.5 billion from hedge funds, according to Hedge Fund Research.

“I’ve been saying for some time that the two-and-20 model is dead,” said Christopher J. Ailman, chief investment officer for the California State Teachers Retirement System, which manages assets of close to $200 billion.

“Take the Pershing Square example,” he said. “Investors are only capturing 28 percent of the gains, which is totally out of whack. If it were my money, I’d say it should be the other way around — the investors should be keeping 70 percent, or even more, like 75 to 80 percent. That’s what I’d consider fair.”

Mr. Ailman said that he and the chief investment officers for several large pension funds were seeking an alternative fee structure that would preserve a performance incentive for managers but more equitably share the risk. “A few very big states are really thinking through this,” he said.

One approach under consideration, he said, is to use a rolling multiyear period for measuring performance fees. In year one, for example, an investor would pay only a portion of the performance fee if there was a gain. If there were losses in subsequent years, the investor would claw back the withheld compensation. That would solve the Pershing Square Holdings problem. “Quite a few large investors are thinking along these lines,” Mr. Ailman told me. “Of course the devil is in the details.”

Such an approach is anathema to most hedge funds, which rely on annual performance fees to compensate their highly paid — some would say overpaid — staffs.

Even so, hedge funds are competing more fiercely over fees. In a nod to investor concerns, earlier this year Mr. Ackman offered investors an option to pay no performance fees on gains of less than 5 percent, but a steeper 30 percent on gains above that level.

“Management fees pretty much used to be 2 percent,” Mr. Ailman said. “Now we’re seeing them as low as 70 basis points.” And performance fees, known as carried interest, have dropped in some cases from 20 percent “to the low teens and even as low as 10 percent. And these are large, well-known funds.”

Hedge fund defenders have said it isn’t fair to pick just four years of performance, like the Pershing Square Holdings example, saying it is too short a track record. Thanks to Mr. Ackman’s early successes, his longtime investors have fared much better than the more recent ones, and cumulative fees are closer to the percentages in the stated fee structures.

In his letter to investors, Mr. Ackman pointed to much better results for his older hedge fund, Pershing Square L.P. Since it started in 2004, it has produced annualized compounded returns after fees of 14.9 percent, more than double the S.&P. 500’s 7.6 percent.

Still, he acknowledged, “while this is a good result, it is below our long-term goals and not much solace” for “investors who joined us in recent years.”

A version of this article appears in print on December 23, 2016, on Page B1 of the New York edition with the headline: Hedge Fund Math: Heads or Tails, They Win. Order Reprints|Today's Paper|Subscribe